How do I make cron run something every "N"th minute, where n % 5 == 1?

1-56/5 * * * * /my/script

This should work on vixiecron, I'm not sure about other implementations.

Use your first schedule:

*/5 * * * * /my/script

And add this to the start of your script:

sleep 60

(Yes, this is a joke)

This is quite an old topic, however as so much time has passed there are a few other options now. One of which is not to use cron at all, and use systemd timers. Using these gives you a higher granularity than seconds along with lots of other options

More information is available here https://wiki.archlinux.org/index.php/Systemd/Timers

eg to run a adhoc command

# systemd-run --on-calendar="*:1/5" /bin/touch /tmp/foo2Running timer as unit run-r31335c4878f24f90b02c8ebed319ca60.timer.Will run service as unit run-r31335c4878f24f90b02c8ebed319ca60.service.# systemctl status run-r31335c4878f24f90b02c8ebed319ca60.timer● run-r31335c4878f24f90b02c8ebed319ca60.timer - /bin/touch /tmp/foo2   Loaded: loadedTransient: yes  Drop-In: /run/systemd/system/run-r31335c4878f24f90b02c8ebed319ca60.timer.d           └─50-Description.conf, 50-OnCalendar.conf, 50-RemainAfterElapse.conf   Active: active (waiting) since Wed 2017-10-25 09:05:13 UTC; 40s ago# ls -l  /tmp/foo*-rw-r--r-- 1 root root 0 Oct 25 09:06 /tmp/foo2# sleep 300; ls -l  /tmp/foo*-rw-r--r-- 1 root root 0 Oct 25 09:11 /tmp/foo2# date; ls -l /tmp/foo2Wed Oct 25 09:21:42 UTC 2017-rw-r--r-- 1 root root 0 Oct 25 09:21 /tmp/foo2

edit: these type of timers wont persist over reboot, if you want them to make sure you generate a proper service file, with the relevant oncalendar line